(i) With the help of a labeled diagram, descried briefly the underlying principle and working of a step up transformer.
(ii) Write any two sources of energy loss in a transformer.
(iii) A step up transformer converts a low input voltage into a high output voltage. Does it violate law of conservation of energy? Explain.
Derive an expression for the impedance of a series LCR circuit connected to an AC supply of variable frequency.
Plot a graph showing variation of current with the frequency of the applied voltage.
Explain briefly how the phenomenon of resonance in the circuit can be used in the tuning mechanism of a radio or a TV set.
(i) Principle underlying the working of transformer: The principle is Mutual Inductance. When a changing source of voltage is introduced across a coil (which is physically coupled to another coil), the changing current through it induces an EMF across the second coil.
A transformer consists of two sets of coils, insulated from each other. They are wound on a soft-iron core, either one on top of the other, or on separate limbs of the core.
One of the coils is called the primary coil, and has Np turns. The other coil, the secondary coil, has Ns turns. The relative numbers depend on whether the voltage needs to be stepped up or stepped down.
By definition, the voltage to be transformed is introduced across the primary coil. When the alternating voltage is applied across the primary, the resulting alternating current through it produces a changing magnetic field, whose flux through the secondary coil changes.
From Faraday's law, this changing flux induces an EMF across the secondary, whose magnitude depends on the amount of coupling of the two coils, numerically measured as mutual inductance. The more this coupling or association of the two coils, the more is mutual inductance, and therefore the induced EMF.
If is the flux through each turn of the core, then through N turns around the core, the total flux is N.
So, the EMF induced in the secondary coil is
Similarly, there will also be an EMF induced in the primary coil itself, due to self inductance, given by
If the voltage applied across the primary is Vp , then if its resistance is R, the current through it will be
However, assuming negligible resistance, since we cannot have an infinite current through the coil, then
If the secondary is an open circuit, no current is drawn from it then, voltage across it will be
From equations, it is clear that
If the transformer is 100% efficient, that is, all the input power is transferred to the secondary without any leakage or losses, then
IP VP =IS VS
This implies that
It is clear from that if NS> NP the voltage will be stepped up, and if NS< NPit will be stepped down.
However, in a step down transformer, there will be a greater current in the secondary as compared to the primary and vice-versa.
(ii) The possible sources of power losses in practical transformers can be:
(1) Flux Leakage: Not all flux of the primary can be associated with the secondary. There is always some flux which due to lack of absolute coupling, can leak. To avoid this, the coils are wound over each other again and again.
(2) Resistance of windings: The transformer coil wires cannot have absolutely zero resistance, so some Joule loss is inevitable.
(3) Core eddy currents: Since the core is a very good conductor itself, currents are induced in it due to changing magnetic fields, called eddy currents. These also result in losses.
(4) Hysteresis: Some part of energy is frozen into the core permanently in the form of a residual magnetic field due to its ferromagnetic character.
(iii) No, it does not violate the energy conservation. When low voltage is converted to high voltage, the current is lowered, thereby conserving the total energy dissipated across the primary and secondary coil.
(i) In the phasor diagram, since at t=0, the external source of EMF is V=V0 (peak value), which is the x component of the phasor V, this vector will be along the X axis. The current phasor I will be at an angle relative to this.
So, since VR=RI it will be parallel to this current phasor, at angle relative to V.
Similarly, looking at equations (0.32), it is evident that VLwill make an angle and VCwill make angle relative to VR.
This also implies that VL and VC will lie in opposite directions, as the following figure shows.
It is obvious from the figure that the vector VR is perpendicular to the vector VL+VC, and also, of course,
V= VR +( VL+VC)
Taking the dot product VV gives
= VR +( VL+VC)
From this, we immediately get
The factor Z is the analog of resistance in a purely resistive circuit, and is called Impedance.
The phase is immediately found from the simple phasor picture
(ii)The current as a function of source frequency is plotted below:
(iii)Whenever one needs a selection mechanism to select a particular frequency out of a range of frequencies, such resonating circuits are useful.
For instance, the tuner in a radio is precisely such a circuit, whose L and C can be varied. Varying these components varies the resonant frequency. As soon as the resonant frequency matches a particular external signal (radio signal) frequency, there is a sharp response, and the device picks up that signal.
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